6 1. Introduction x y θ r A B R R γ Figure 1.3. Path through the earth m that moves between two points, A = (ra,θa) and B = (rb,θb), on or near the surface of the earth. We now wish to find the planar curve γ that minimizes the travel time T = T 0 dt = γ dt ds ds = γ 1 v ds = γ 1 v dr2 + r2 dθ2 (1.8) between A and B, where s is arc length and v is the speed of the particle. When a particle is outside a uniform spherical shell, the shell exerts a gravitational force equal to that of an identical point mass at the center of the shell. A particle inside the shell feels no force (see Exercise 1.6.3). By integrating over spherical shells of different radii (Exercise 1.6.4), one can show that the gravitational potential energy within a spherical and homogeneous earth can be written V (r) = 1 2 mg R r2 , (1.9)
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